If the funding source is the federal government, you may not be eligible.
You have to be very careful about this because a few consortium sites will say that citizenship is NOT a requirement. For example, San Diego VA, New Mexico VA, South Carolina VA, and I think Mississippi VA have positions that are half-VA and half non-VA (an affiliated university, for instance). Of the positions within the consortium, you may apply to those that are not 100% VA. Unfortunately, sometimes that means you cannot apply to a position that is within your specialty area.
In other CMHC and university hospitals, the same thing may happen. If a position is funded by a federal/national grant, you may not be considered. So, even when the APPIC directory states that citizenship is not required, you may still be deemed "ineligible" eventually.
The Canadian sites have similar issues. When they tell you Canadian citizenship is not required, it can mean a number of things. In general, there is a law that says they must hire citizens first. Also, helping you get a work visa can be a hassle and many sites will be reluctant to do so.