I plan to use a solid state relay to control a couple of small 12 VDC motors. One is 12V, 12Watt (1A) and the other is 12V 11amp peak. The SSR can handle 15A DC. Some of my inputs (different part of the device) also use the same 12VDC power supply. I am using a SSR and the output already has a diode. Should I still include back EMF protection? Should I use a diode? Zener diode? Varistor? I am concerned about back EMF but also a possible drop in voltage that might cause the inputs to false read. Here is the SSR diagram.
Yes, you should put a diode across the load, just as shown in the SSR diagram.
Otherwise you can damage the SSR from the inductive energy in the motor when the SSR tries to turn off.
The diode can be a conventional silicon rectifier- it will have to conduct the motor current briefly and needs to withstand the supply voltage plus some margin. For the 1A one, a 1N4004 or 1N5404 will work fine.
For the 11A one, a diode rated at 10A or so should be fine, such as a 10A04-T (or you could use that type of diode for both).