Miss Maudie does not chew gum.
One day, Scout makes a discovery that for a child in the thirties would have been wondrous. This discovery is made in the knot-hole of a tree on the edge of the Radley property.
Some tinfoil was sticking in a knot-hole just above my eye level, winking at me in the afternoon sun. I stood on tiptoe, hastily looked around once more, reached into the hole, and withdrew two pieces of chewing gum minus their outer wrappers. (Ch. 4)
When Scout finds the gum in the tree, the reason that Jem does not want her to chew it is because he is not sure where it is from and whose it is. Remember that it did not have the wrapper. Scout assures him that she has been chewing it all day and did not die.
The children find a fresh package of chewing gum the next time, with all the gum still in the wrapper. They decide to write a letter to the owner of the gum, thanking him or her for giving it to them. However, they are not sure who is leaving the presents in the tree. When Scout suggests that it was Miss Maudie, Jem says it couldn’t be, because she doesn’t chew gum.
“Ar-r, Miss Maudie can’t chew gum—” Jem broke into a grin. “You know, she can talk real pretty sometimes. One time I asked her to have a chew and she said no thanks, that—chewing gum cleaved to her palate and rendered her speechless,” said Jem carefully. “ (Ch. 7)
What Miss Maudie is saying is that when she chews gum, it makes it hard for her to appreciate the taste of anything else. Therefore, it is not that she can't chew gum, she just doesn't like to chew it, because it sticks to her mouth.
Eventually, the children realize that the tree is Boo Radley’s. Nathan Radley cements the knot-hole, claiming that the tree is sick. He does not want Boo communicating with the children. They are saddened by this. They have spent all of the summer trying to get Boo to come out, and now they have a way to connect with him.
The leaving of presents in the tree by Boo is an overture of friendship to Scout and Jem. He leaves them little trinkets that only have value to children. The fact that soap dolls, pennies, twine, and gum mean something to him shows that he either has the mind of a child or understands children. Nathan Radley clearly wants nothing to do with this. He tries to prevent the interaction either because he doesn’t understand it or because he feels it can lead to nothing good.