Removing and using programmed Atmega 328 of Arduino UNO on customised PCB
Yes, it can. Your board will need to provide +5v and ground, a reset signal, and a crystal for the chip's programmed clock frequency (unless you reprogrammed it to use its internal clock). The power and clock signals will need filter capacitors. You can even re-program it on the board with an FTDI (USB-TTL Serial) cable. 

There are lots do-it-yourself instructions to build an Arduino board, which is a good place to start, even if what you eventually want is a larger system board with an Arduino integrated into it. This is one I can vouch for; I built Mr. Alves' Version-1 as my only Uno-like board and have been using it for several years.