Mjayadev, all modern DSLR cameras use IR cutoff filters that are, in the grand scheme of things, quite cheap. Even the 60Da, which improves Hydrogen-alpha transmission, still barely achieves 70% transmission at that band (656nm). Most standard DSLRs have an IR cutoff filter with a long shoulder (or heel, rather) that gradually reduces deep reds and infrared transmission up through around 1100nm. Beyond 750nm, transmission of the near-IR band is low enough that it is largely inconsequential.
With the 1D X, I would expect Ha (656nm) transmission to be somewhat poor. It won't really be any better or worse than other DSLRs though, however it is not as good as the 60Da.
That said, should you worry? It ultimately depends on what you want to do. Are you just interested in photographing ultra wide field milky way images with landscape foregrounds? If so, don't EVEN worry yourself about the IR cut filter.
If you want to do standard wide field imaging (anywhere from a few degrees to tens of degrees of sky), and your primary goal is full color imaging, I still wouldn't worry about Ha. Your images will be colorful, but not necessarily "realistic" as they will lack richness those deep red tones (which include Ha, S-II (Sulfur II) @ 645nm, and N-I & N-II (Nitrogen I & II) @ ~650nm and 658nm). You could also do narrow field astrophotography for objects less than a few degrees in size, but again, you would lack the richness of those deeper red tones.
Overall, if you are generally just starting out with astrophotography, there are probably many more things you should be concerned with than what bands your sensor is sensitive to. There is a lot of technique you will need to learn in order to image the night sky effectively, especially if you are doing wide and/or narrow field imaging of DSOs. For any of these objects, you will need long "total integration time", which basically means total exposure time, via stacking multiple images, in the realm of 10 minutes to many hours long (and for some objects, exposures might need to span multiple nights!)
In order to achieve that, you will need a tracking mount. More specifically, you'll need a german equatorial tracking mount. Tracking mounts come in a very broad range of costs and capabilities, ranging anywhere from a few hundred bucks for a basic mount capable of holding up to around 20lb, and capable of tracking up to a few minutes at most; through midrange mounts costing anywhere from a thousand to several thousand capable of holding up to around 40-60lb and capable of tracking up to five minutes, and maybe 10 minutes with very careful polar alignment and the assistance of an autoguiding setup; through high end mounts costing tens of thousands capable of holding anywhere from 45lb up to many hundreds of pounds, and capable of tracking anywhere from 10 minutes through many tens of minutes (30 minutes and longer) with exctremely high precision.
The minimum recommended type of mount for entry level and midrange astrophotography are the Celestron CGEM & CGEM DX, Orion Atlas EQ-G, Skywatcher EQ-6, and iOptron iEQ, ZEQ, and CEM lines. These cost in the range of $1400 through $3500, and when properly aligned with the actual celestial pole (north or south) will get you fairly good tracking up to around 5 minutes (300 seconds). With the added support of a guiding scope and automatic guide camera hooked into a laptop computer, this can get you up to maybe 10 minute (600 second) exposures. This is a far more important aspect of astrophotography, especially early on, than whether your IR cut filter transmits enough of the deep red bands.
To get good color fidelity, luminance detail, and maintain sharp (vs. oblong or trailed) stars, you will need tracking good enough to support a minimum of 4 minute (240 second) exposures, and a maximum of 10 minute (600 second) exposures. A good CGEM or Atlas mount and autoguiding is pretty essential for that. Once you have all of that taken care of, then you can start to concern yourself about other things, like emission bands and maybe even LRGB or narrow band (SII, Ha, OIII) filtration and mapped color (especially if you live in a city, where light pollution is going to limit your exposures and reduce your detail anyway...narrow band imaging can be a godsend, albeit a more complicated and meticulous form of astrophotography.)
Achieving exposures longer than around four minutes requires very good skill with polar aligning your mount, and with drift aligning your scope. Polar alignment can take some time to get precise, although setup routines in the midrange CGEM, Atlas, EQ6 and iOptron mounts simplify it a bit. Drift alignment is more important, in that it reduces declination drift. Equatorial mounts track stars across the sky, without field rotation, by tracking in right ascension. If the declination axis is not aligned properly, stars will "drift in declination", and over time that will result in oblong stars and soft nebula detail. Declination drift requires some good skill to refine to the point where you can expose for more than a few minutes.
Even once you have aligned your telescope for smooth declination when tracking, periodic errors in the mount's gearing will still introduce some periodic wobble that will again reduce the accuracy of how stars are imaged. This is where autoguiding comes into play...autoguiding uses a small webcam like video sensor attached to a separate small guiding scope, along with some software (like PHD, or Push Here Dummy) to lock onto a star and make sure it tracks well. When a star's tracking deviates from where it should track, the the autoguider will instruct the mount to either speed up or slow down in RA, or additionally correct in Dec, to maintain tracking accuracy. This can get you up to 10 minute exposures when you have everything precisely aligned.
Finally, exposing the night sky to good, rich color and low noise usually requires very long total exposure times. For deep sky objects (nebula, galaxies, clusters, etc.) you will usually need to expose multiple 4-10 minute exposures of the same exact region of the sky while the telescope tracks. You'll probably also need to produce dark frames and bias frames. All of this is then registered, calibrated, and stacked in a tool like DeepSkyStacker, to produce a final "integrated" image that has much less noise and much richer detail and color fidelity, than one single exposure.