Based on some of the cases mentioned in some news media, maybe some are "sue-happy". Then again, there are times when a person may have more than enough solid and legitimate reason to sue (whether anyone else is aware of, or oblivious to, that reason or not). Not suing under legitimate circumstances is just stupid and/or spineless. I don't think there's any "non-Christian-like" about addressing an injustice or failure on someone's part through legal, civil, and peaceful means.
In fact, whether it's something like a violation of one's rights or someone else's carelessness and/or stupidity, maybe it's more "Christian-like" to sue and make sure some people learn a thing or two (so maybe they won't pull the same crap again) than just to go the spineless/cowardly route of "happily" accepting someone else's screw-ups, as well as paying the price for them.
You can be a person who pretty much does everything right, or at least tries to; and you can be someone who is more than able and willing and objective enough to see some things that for which you are responsible and acknowledge that.
Once I heard on some "news thing" that people who have been the victims of medical mistakes are less likely to sue when the medical facility/professional recognizes his/her fault and apologizes. It's when they won't own up to a screw-up that people then are more likely to sue. I don't imagine it's much different in other areas of life either. In other words, most people who have some "issue" foisted on them through no fault of their own are willing to own up to their own mistakes and expect others to do the same.
Suing is a civilized way of aiming to get some form of justice, aiming to hold people accountable, aiming to teach any number of lessons about why you can't mistreat other people, and (ideally) making sure that all involved (and some others too, maybe) get to the facts/truth when a guilty party refuses to own up.
I see nothing "un-Christian-like" about any of that UNLESS an attorney involved tries to win at all costs without regard for truth/justice and with regard only for winning AT ALL COSTS (and that, to me, goes for all other parties involved in a case). Some people/attorneys are happy enough to win at all costs and without regard for anyone/anything else. That, to me, is when it becomes "un-Christian-lke:", shameful, sleazy, cowardly, weak, and even immoral.
Believing someone is "sue-happy" is likely the result of not knowing the facts of a case.