What causes turning ON a single output pin on Microchip PIC16F690 to spontaneously turn OFF another pin on the same port?
It's the well-known read-modify-write problem, you will find details in the data sheet. You have to write to the whole register, as you have found. The 18F and 16-bit devices don't have the problem. There is a good description on page 2 in this document. Changing bits in a "shadow" register then writing the register to the output port is often used to get round the problem.