Sorry to hear about your MIL 
I think you are right to think carefully about this and the immediate aftermath of a major bereavement is not usually the right time for anyone to make life changing decisions. Sure, have your FIL to stay on an extended visit but I think the implications of a major move need a really open and honest conversation - future care needs come into this but also there are a lot of more immediate considerations, e.g. what will happen to his house and possessions, what about inheritance issues, what do any other siblings think about it? What about your own parents, will they feel put out not to have been invited to move in too, what about your kids, after all it's their home too. What are your FIL expectations re housework, cooking, contributions to household bills, socialization etc. E.g. would he expect to live a relatively independent life but within your house e.g. will he come and go as he pleases under his own steam, cook his own meals, sit and watch TV or whatever on his own of an evening or will he expect 'full service' i.e. to eat 3 meals a day with the family, have lifts anywhere he wants to go, be entertained or at least with you every evening? What about if he needs adaptations to the house (grab rails, stair lift etc) - are you happy for these to be put in and who will pay?
Contrary to popular MN opinion I have seen multi-generational living work really, really well but it totally depends on good communication and mutually understood and agreeable expectations on how it will all work both financially and practically/emotionally.
To answer your actual question, what do you mean by 'eligible'? Do you mean eligible for state funded care? Pretty much anyone can choose to check themselves into a care home if they want to regardless of true 'need' or eligibility (most care homes do not have a problem taking someone who's needs are less than the home can manage, the problem can be the other way) if (and it's a big if!) they are paying the bill themselves and they are going of their own free will (or if they have lost capacity to make their own decisions due to e.g. they have executed a power of attorney and their nominated person agrees to them going into a home). So like I say above, in the scenario where there's plenty of money so finances aren't a problem (and bear in mind care homes can be staggeringly expensive), you'd need to understand from FIL whether he agrees he will only live at your house so long as his care needs are within what you think you can reasonably manage (and by who/how will it be decided when this point has passed) and that after that he will willingly go into a care home (and he will execute a power of attorney in favour of DP so that decision can be made for him if he's no longer capable of deciding for himself) AND your DP and any siblings are 100% on board with this and aren't going to kick up rough about "their inheritance" being spent on care home fees in the expectation you'll act as carer.
If money is an issue it becomes somewhat more complicated - the state will only pay for a care home as a last resort, i.e. when it is assessed by social workers and/or medical professionals that they can't live safely at home with carer assistance. This is quite a high bar, and of course often if there's resistance from the person themselves to going into a home (which is not uncommon) there can be a distinct inclination to leave them be until a crisis happens. There's no particular reason why this would be a different process if your FIL is living with you than if it was with a spouse or on his own, no-one can be forced by law to provide care against their will BUT I think it's unrealistic to expect that you could simply say to SS you're not providing any care and he'd then be whisked straight off to a home no questions asked. In reality there would likely be as a minimum a degree of pressure on you and DP to provide at least some care (especially if you are saying you won't accept carers coming in) both from social workers but also FIL himself, siblings/wider family, and your own consciences, when it comes to it I doubt many people would honestly allow a relative living in their house to not be able to dress, not wash, not get to the toilet in time, not have proper meals, be at risk of falls etc out of a stubborn desire to get social services/the person to agree to a care home - and then of course it can be a vicious circle, the more you help the better the person is coping and the more it seems to SS who that there's no additional care needs etc etc. I honestly think you need to think carefully through what risk you are prepared to take here, it is a risk not a given of course, plenty of people manage fine right to the end of their lives without needing any significant/personal care so the question may never arise but you do need to have these conversations, with FIL and with DP I think.
Good luck!