I searched the rules forum on this but it wasn't answered.
You need at least two hands to climb, but you can cling to a wall with one hand while you cast a spell, shoot a small arm, or take some other action that takes only one hand.
But Climbing Suckers is an augment that attaches strictly to your feet and gives you a climb speed. It mentions nothing about using your hands.
The soles of your feet become covered in fleshy slats you can open and close, providing you a climb speed of 20 feet (seepage 259). You don't need to attempt Athletics checks to traverse a vertical or horizontal surface (even upside down). You don't benefit from climbing suckers unless you're barefoot or wearing custom clothing, and you can not be wearing more than light armor. Climbing suckers integrate into your natural feet and do not require you to replace your feet and legs. Climbing suckers can be installed into the feet of prosthetic legs
Even the page direction about Climb Speed on pg. 259 mentions nothing about using hands, in fact it states if you don't have a climb speed it says you use the Athletics skill to climb.
If you have a climb speed, you can use move actions to climb slopes, walls, and other steep inclines, and you don't need to attempt an Athletics check to climb except in hazardous circumstances (see the Athletics skill starting on page 136). You are not flat-footed (see page 276) while climbing. You receive a +8 bonus to all Athletics checks to climb and can always take 10 while climbing, even if distracted or threatened. You can move your full climb speed when you use the move action while climbing, but you cannot run. You can move double your climb speed with a successful Athletics check to climb, but you take a -5 penalty to the check. Creatures without a climb speed use the Athletics skill to climb.
So do you need hands to climb with Climbing Suckers? (my opinion is no)
My thought is also "no, you do not need your hands in this case."