Planarity conditions for Planar 1-in-3 SAT
Yes you can. Actually you can even show that something stronger is true. The problem know as Positive Planar 1-in-3-SAT is NP-complete as shown by Mulzer and Rote. 

In this version of 1-in-3-SAT, you require for every input formula that


you have three variables per clause, none of them negated
the graph of the formula is planar, even if you add the "backbone" between the variable vertices


The reduction is from Planar 3-SAT.