We have specific school positions on our Board: Principal, Vice Principal, and 2 Teacher Reps.
Our bylaws state that:
Nominees must be PTO members in good standing and may not be faculty/staff of {school name} or hold a management position with the school district.
Note - we have a large board of 25-30 people with all the officers, reps, and commmittee chairs. Only the 5 officers are elected and subject to that clause.
As far as experience, yes you can put that clause in, but you may want a safeguard:
1) either it's that the experience is "preferred" rather than required, which gives it no teeth but expresses the intent, or
2) make it a requirement, BUT allow yourself an override option.
Our group follows #1, but this passage also provides a couple examples of override options:
No officer shall hold the same office for more than two consecutive years, unless approved by a 2/3 vote of the Executive Board. Candidates for presidency should have prior experience on the Executive Board. Two members residing in the same household may not simultaneously serve on the Executive Board unless expressly approved by a majority vote of the Executive Board.