Could there ever be a 'CzeXit' over this?
I'll take the liberty and go into more explaining when it comes to this question.
Czechs have been the most Eurosceptic nation during the "good times", i.e. before the Euroe crisis. We fared relatively well during the crisis so while the mood here hasn't changed it did change drastically in other countries so while couple of years ago the Czech Euroscepticism would be considered extreme, by today's standard in countries like UK, Greece or Hungary, it is pretty mild.
The guns themselves won't be a deal breaker leading to Czechixt simply because they don't affect enough people. It affects about 3% of population directly and about 10% if we extend individual licenses to family circle. On the other hand, these are all folk who take the issue personally and who are deciding actively as regards their political choices (we don't have any vocal antis in the country, so there are no votes to win by being anti-gun rights, only votes to lose). The whole EU gun ban has brought the community together though. I personally used to be pretty pro-EU before this, and now I think you can guess what my attitude is. This is the same for basically everyone of this 10% of electorate.
In normal circumstances, EU gun ban would be an issue only for the 10%. Czech Republic is extremely safe country by Western European standards, and probably unimaginably safe by US standards. So most of population don't care about guns, and they don't really realize they do have A RIGHT to carry a firearm (subject to getting a license first) until they find themselves in situation when they do need to enjoy that right (which is unlikely to happen).
The migration crisis has been completely absent in the Czech Republic thanks to our combination of restrictive immigration laws (you don't get asylum here, only "international protection", i.e. you either
earn your citizenship, or get sent back once your home country is safe - immediately and forcibly) and low welfare (you won't go hungry, but that's about it). Yet as absent as it may have been in the streets, it has been all over the news and just behind the border in Austria and Germany. It is 3 hours ride with train to Vienna or Munich from Prague, and there is no border check in between. This has made many people start thinking what would they do should Cologne come here. And that's when they realized they have the RIGHT. Even if they didn't go and run to get a license (many did, more didn't).
And now EU wants to take this RIGHT they even didn't enjoy yet away from them. So there is unknown % of people who take this seriously over the 10% we already know take it as deal breaker of Czech EU membership.
Then there are those for whom EU gun ban is not an issue but who are still unbelievably angry about the immigration mess, and especially about the forced quotas - Czech Republic voted against redistribution of migrants, and the majority voting was actually the first time a decision was taken against a country's will in the EU Council. We are especially sensible to that since the Munich Agreement. So this adds unknown % over the 10% and X%.
So far, the forced distribution affects only 160.000 migrants, of which only 4.000 should come our way. However, there is "permanent quota redistribution" on the table. If that was passed against our will (and there is no way in hell any Czech government would accept it), then I would say yes, next election would be clearly won by any party offering Czexit.
Meanwhile you need to keep in mind that Czech Republic has the lowest unemployment in the EU, low crime, good healthcare, nice standard of living, no migrants to stirr things up so far - so in general things are going well and people don't tend to like huge changes like leaving EU when things are working out for them.
Interestingly enough, Slovakia is actually the only country that took EU to the EU court over the 160K redistribution (and that is EXTREMELY hostile act in EU politics which happens about once in a decade, and never by a small players like Slovakia). Which means that it is extremely unlikely that the permanent quota will be put to vote now that Slovakia will take EU presidency in Q2 2016.
Czexit may be a possibility in a couple of years, but not because of the guns only. Meanwhile our government will be finding ways to flex whatever EU gun legislation is coming our way if not for any other reason than because they don't want hundreds of thousands of legal guns entering the black market.
They actually made the point out loud that while today terrorists use beaten up unreliable guns that lied in post-soviet/yugoslav stock for decades before being smuggled to the EU, if the gun ban is passed then the black market will be full of currently legal tricked-out vz.58s and brand new AR-15s. Simply because there is no chance in hell owners will just give them up.