If Green Card was swiped, if that is what and how this happened, unless this happened this month or at least very recently, the odds of actually getting the data changed are probably nil. Even a full-blown, time-consuming, and expensive lawsuit has little . . . well, really, probably zero chance. After all, the record has no direct impact in and of itself. It is not a record which directly affects status.
There may be a process, perhaps a part of what is available through the Traveler Redress Inquiry Program, for having the record supplemented by some kind of statement, a note or flag which in effect shows the data is contested, that is that the accuracy of the record of entry into the U.S. on that date is challenged, with an explanation (why, how, information relevant to the credibility of the challenge). For what that would be worth, if that is available.
Note, too, that even if "corrected," that correction itself would likely be flagged as such, with an explanation. So either way, the record will probably continue to show an event on that date, either as corrected with flag explaining correction, or as challenged with explanation for challenge.
What IRCC will see, then, if and when IRCC accesses the database maintained by CBSA, to generate a report of travel history, is hard to forecast.
There are of course implications regarding status in the U.S. to consider, since there are residency obligations attendant the GC as well.
Proof of actual presence in Canada after that date (the sooner after the better) will at least be evidence of presence in Canada after that date, and together with other evidence of life being lived in Canada after that date, that should suffice to prove presence in Canada IF, IF such proof is requested (CIT 0520 or CIT 0171, that is RQ-lite or full-blown RQ). Whether that happens or not depends on many other factors, and of course a review of the CBSA travel history may indeed be what triggers such non-routine processing. Remember, many people put way, way too much emphasis on exit and entry dates. Presence in-between such dates can be as much in issue as the travel dates themselves. In your situation, for example, proof that you were at work in Canada or visited a doctor or some other activity overtly showing your presence in Canada, will prove your presence in Canada, at least as of that date.
How long ago this occurred is probably a significant factor. Prompt efforts to correct data-entry records are typically more credible than stale efforts.
On the other hand, having a U.S. GC in and of itself (which the citizenship applicant must disclose in the application, or notify IRCC if GC obtained while the application is in process) can be a big factor in triggering non-routine processing, including the full-blown RQ. The risk of this, like most risks, depends on many factors and circumstances, including when the GC was obtained. In and of itself, a U.S. GC evidences strong residential ties in the U.S. and virtually mandates IRCC make more probing inquiries to verify the applicant's physical presence in Canada. Which, in thinking about this aspect of things, is indeed an indicator that IRCC will be more likely to examine such an applicant's travel history and travel to the U.S. in particular.
Which is to acknowledge that there probably is an elevated risk this will be looked at and considered, and thus will potentially if not likely trigger at least some CIT 0520 requests.
Obviously this imbroglio will influence how much of a margin over the minimum presence you wait to have, that is how long you will wait before making the citizenship application beyond reaching the minimum, if you have not yet made an application. Definitely a good idea to take into consideration the influence of having a GC and what objective documentation you have to prove your presence in Canada after this purported exit date. A much bigger margin than most would probably be prudent. (I had very different, but nonetheless some salient concerns, when I applied for citizenship, so I actually waited nearly two extra years beyond when I met the actual physical presence threshold . . . few need to do that, but some of us are indeed better off waiting much longer to apply, for various reasons. Perhaps one of the more common errors prospective applicants make is to rush making the application based on when they pass the minimum presence threshold.There is much, much more to consider.)
If you have already applied, then your task will be to gather strong evidence of your presence, evidence for all the time you declared being in Canada, and especially objective documentation of presence after the date of this purported exit. (Note: many other circumstances will factor into how things like this influence the process; for example, there can be elevated skepticism when assessing a PR whose spouse is working in the U.S., for obvious reasons. So it is important to not be too narrowly focused when gathering evidence and proof, to be sure to document all periods of presence in Canada.)
You may never be required to actually prove your case to that degree. Very few applicants are. (These days. During the Harper era, in contrast, there were times when the RQ rate went higher than one in every four applicants.) But you are now aware of something which could throw IRCC off the routine rails, so you are aware that it would be prudent to prepare for this:
-- wait longer to apply if you have not yet applied, and also focus on gathering more proof than the ordinary applicant might
-- focus on gathering proof of presence, of a life being lived in Canada, if you have already applied
