Can I use an "8 speed compatible" crank such as Shimano FC-2303 with a 7 speed Shimano MF-HG37 freewheel?
The FC-2303 is a non series road crank set. It will work, with some shifting degradation, but it is not an ideal match. 

8 speed road shifters, and 8 speed mountain shifters pull a different amount of cable, because the spacing of the chain rings on a road crank is different than that on a mountain crank set. 

If you want it to work well, you need to also change your front derailleur, and front shifter. Depending on your bike and the direction of cable pull, you may or may not be able to put a road dérailleur on it, but you can get a touring dérailleur. And you'll need an 8 speed flat bar road shifter which will pull the correct amount of cable for for an 8 speed road crank set.