Sorry to answer a questiong with a question, but are they limiting volunteers to American Citizens only?
There are plenty of ways to verify a person's background without a SSN to the county. Driver's License numbers work quite well. The state already has the SSN on file associated with the DL#.
To continue the trend of a question for a question
What about people who don't drive cars? Also what you propose is one agency sharing your info with another without your permission.
The short version of this is you can either provide the requested information or contact Kane Co to see if they will allow alternative methods. At the end of the day the land manager makes the rules we have to adbide by.