There is no way they're getting through 1/4 inch mesh. That's plenty small enough.

If you leave any gaps though they'll find them, or pry them open bigger. 1/2 inch is like a highway to them.

Something you may not have thought of: You're probably using several different metals, outside in a moist atmosphere. There may be steel nails, aluminum screen, copper pipes and wire, metal siding, etc. Any time two different metals are connected you have the potential (pun) for galvanic corrosion. You can eat a hole surprisingly fast.

http://corrosion.ksc.nasa.gov/galcorr.htm