Can I wire a GFCI outlet with 3 hots and 3 neutrals?
I thought I should expand on my comment and others about box fill.
With 3 hots 3 neutrals and at least 1 ground (I hope) using 14 gauge wire the calculation would be 7 conductors x 2 cubic inches + 2 more if there are clamps in the box. The device requires 4 cu inch so your box would have to have a a total 18 cu inch or 20 cu inch if it has clamps. NEC table 314.16.B.
 The largest listed box per table 314.16.A (single gang) is 3 X 2 X 3-1/2 with a volume of 18 cu inch. if you have 1 12 gauge wire or clamps in the box your plan would violate the box fill criteria for a single gang box.

#14 wire each conductor counts as 2.  #12 each conductor counts as 2.25 you only have to count 1 ground wire (no matter how many are in the box but the largest size must be used). 
The device is 2X the largest wire size connected to it. 1X for clamps based on the largest wire size. 

The last thing is the box volume must be stamped on the box or the volume from table 314.16.A used.  Measuring the box will not give the proper volume per code requirements. For example the 3 x 2 x 3-1\2 box would calculate at 21 cubic inches but code only allows 18 cubic inches for the box fill calculation. 

I hope this helps clarify what is being said about box fill. some GFCI's do run warm and stuffing them in a over filled box can lead to a wiring and or device failure.