Not really, sorta, except when.
If you have just normal mosquito levels, it is not generally a problem. Windowscreen on the windows/vents does decrease airflow noticeably so *if* your coop is borderline not really quite amply vented to begin with it may be best to leave it off (well, really "best" to cut more vents
)
However, mosquitos can transmit some diseases (west nile, avian pox, probably other htings I'm not thinking of at the moment), also *huge horror-movie style hordes* of mosquitoes can be not so good for chickens' health or happiness. So if you live in a mosquito swamp of truly Biblical proportions, screening vents/windows probably IS worthwhile -- there will still be a substantial number of skeeters in the coop b/c they will go in thru the popdoor when it's open during the daytime but it will not be anything like as bad as the nighttime situation *without* screens. The same goes for regions with, what is it, buffalo gnats? that can actually kill chickens.
The difficulty with the "it's extra protein" theory is that while yes chickens will eat some mosquitoes by day, they do not eat in their sleep
and nighttime is when the majority of mosquito-feeding problems occur.
JME living in a mosquito swamp of nearly but not quite Biblical proportions (translation: at least it is not Manitoba
),
Pat