I headcanon Mikasa as being on the spectrum, so in my mind, she tends to get hyper focused on projects (often that involve helping others), and forgets/neglects to eat and sleep. Jean, being the wonderful boyfriend he is, would make her meals and bring her snacks during those times. He'd also do his best to gently pry her away from something, especially if she'd been at it for hours on end, and bring her to bed.
She'd probably be adorably reluctant about it at first, but the further along they got in their relationship, the easier it would be for her to let Jean take care of her.
As difficult as it is for her to accept sometimes, she deserves to be loved and cared for just as much as her friends.