So. I'm very confused by the situation involving my husband right now, so I'm hoping you guys can help me.
Essentially, two months ago, my husband accidentally marked the wrong work as being completed and when he realized he'd marked on the wrong space or box or whatever it is in the logs that he marks off the work from, he didn't inform anyone. He figured he'd go back later that day and just do that work. Well, one thing lead to another and that never happened. Two months rolled by without the work actually getting done. Before he could get the work done, his chief found the logs and questioned him. His chief doesn't believe it was an accident, and he is now being sent to captain's mast for an integrity violation, because they believe he did it on purpose.
First, I hate to be the bearer of bad news, but your husband is in trouble because he simply could have put a straight line through the log and initialed it. That is how people 'cross-out' erroneous entries in logs. I'm not sure why he decided that this wasn't an option, but that's why he's in trouble.
Now, onto your questions
1. If they kick him out...
That usually requires a Court Martial or several Captain's Masts.
EDIT: The CO can't directly kick him out at mast; instead, if the CO recommends removal of the nuclear NEC (which will be granted), then your husband can try to serve as a conventional EM/ET/MM. If the rating is full and the Navy has no other place for him, then he'll be administratively separated within 10 days.
2. I know that integrity violations are big deals for nukes, so if they DON'T kick him out, will he still be a nuke?
Entirely up to the CO's discretion at Captain's Mast. Your husband is going to have a better explanation than what you told me, because the 'no really I totally marked this off by accident and meant to do this maintenance later' excuse smells like BS. Like I said above, he could have just crossed it out and initialled the erroneous entry. The CO also knows this.
At a minimum, he's going to lose his stripes and get docked pay. If he gets de-nuked, he will also have to repay any bonuses he's collected.
3. How common is it for an infraction like this to be taken THIS seriously?
Very common. Your husband works on old but complex equipment that needs to be properly maintained to keep functioning. A failure in equipment due to poor maintenance could have drastic consequences when talking about nuclear power, and at a minimum would cost the Navy millions in corrective repairs.
4. What kind of discharge will they give him for an integrity violation?
EDIT It's not an automatic discharge as described in point 1. If there's no home for him in the Navy, he gets an administrative separation, but I'm not sure what the classification (honorable, general, OTH) for separation is for this kind of thing.
4a. If it's a less than honorable discharge, will he be able to find work?
He won't get discharged, but an OTH would make it difficult to find work. Every employer asks if you have been honorably discharged; they don't ask if you have been dishonorably discharged.