Can I use the regular Shimano 2300 shifter/brake levers with Shimano 105 callipers?
Shimano is great about publishing specification and compatibility charts for their cycling products. They can all be found here including PDF files of previous year's information.  

The latest doc that has info on 2300 series Claris shift levers and brakes is here. In section 4. Brake systems interchangeability we see that ST-2300 brake/shifter units are not strictly compatible with BR-6700 Ultegra or BR-5700 105 calipers (and I assume that means the same for 6800, 5800, R8000 or R700 calipers as well).  

However, we see that ST-6700 and ST-5700 are compatible (with a caveat of reduced power) with brakes used in 2300 series groupsets, so you might be able to get away with using R-6700 or BR-5700 calipers.

If they don't work together keep the original calipers until you swap shifters. Use the compatibility charts to make sure shifters and derailleurs will work together when you consider upgrading those.