Eggshell is always the nice look to go on any room 
Semi-gloss paint is best used in bathrooms.
Eggshell, aka satin, is ok for bathrooms, kitchens, and any other areas where there is a high likelihood of a scuff mark or something getting on the finish. You can wipe if off with a damp cloth as long as you don’t overdo it. It is easier to touch up than semi-gloss.
Semi-gloss is even better, especially in more moist areas, such as in smaller bathrooms that have less ventilation. However, it is definitely a bit more shiny in appearance, more difficult to apply without noticing imperfections underneath, and it is harder to touch-up without it being obvious that it was touched up. If you are dealing with a newly sheet rocked room, you should take extra time to make sure that the walls are super smooth before applying any primer or paint. If you do that, then the semi-gloss will look awesome. We like to put semi-gloss on areas under chair rails too.
Some of this is simply personal preference.
As a side note, ALWAYS make sure you have a good seal around the top of your shower surround so that moisture cannot get under the paint, and if there is no shower door, I recommend purchasing splash guards and shower curtain retainers that have suction cups to keep the inner curtain liner where it needs to be. Also, make sure to have REALLY good ventilation. If the room is small, you should actually get a vent fan that has more power. As a landlord (my other business), this is a constant battle. Or just tile everything around it, problem solved.
I recommend Semi-Gloss or gloss to my customers to best moisture resistance in bathrooms with showers… I also INSIST on an exhaust fan with at least 180 CFM in any bathroom I’m going to install for them.
That’ll depend on the size of your bathroom and ventilation. Large master bathrooms can get away with flat or satin paints given high ceilings, high CFM exhaust vent and nearby windows. If it’s a standard 40 - 50 sqft bathroom the least we recommend is eggshell if you don’t like the sheen on the the semi-gloss, but if there isn’t proper ventilation then semi-gloss is the way to go. You can run a visual test of your existing paint, does moisture tend to smear along the walls near the shower area or in the room in general? Flat finish walls will absorb, unless previously properly sealed and can also tend to stain with moisture. Another trick, about an hour after you shower, stick a piece of tape to the ceiling and if it doesn’t stick there’s not enough air flow, this means semi-gloss would be best, after a really good prep work done to the existing walls and ceiling otherwise nothing will stick and you’ll have bubbles popping out overtime.
For durability semi gloss will work best however it will give a shiny look, we recommend to go with eggshell as long as you have good ventilation.
For bathrooms, we like the look of satin but prefer the durability of semi-gloss.
Satin is more durable and shinier than eggshell and is ideal for the bathroom