Because the bit cheeks come into such close contact with the horse's mouth, especially when a lot of saliva is produced, using a lubricant that isn't food-grade is a risk. Thankfully, there is an inexpensive food-grade solution that should help in most cases, and you likely already have it in your feed room or kitchen.
The simplest solution is to place a drop of vegetable oil (I have had great success with canola oil, but other common vegetable oils such as corn oil, soybean oil, etc., should also work) on each gap in the hinge. The oil will spread quickly, so it's important to move the hinge back and forth to work each drop of oil in before it drips off the bit. Repeat this for each gap, and then hang up the bridle to allow the oil some time to spread through the entire hinge before riding.
If this method doesn't stop the squeaking, it might be necessary to leave the bit to soak in a dish of your chosen vegetable oil so that the oil can penetrate more deeply.
If your bit resumes squeaking after some time, simply repeat the oiling process as needed.