In everyday English, you can use “the” before "chickenpox". However, we usually omit the article in formal contexts.
She’s never had the chickenpox.
Have you ever had the chickenpox?
However, we tend to drop the article "the" in formal contexts.
It is uncommon for adults to have chickenpox.
The best way to prevent chickenpox is getting the chickenpox vaccine.
Touching fluid from the shingles rash can cause chickenpox.
Because chickenpox is an uncountable noun, you cannot use the indefinite article “a”.
My child has the chickenpox.
My child has a chickenpox.
And never use a determiner (the, this, that, my, your, etc.) when using this term in a general sense
Chickenpox can lead to severe complications.
As an adjective, "chickenpox" can be preceded by the article a/an (or another determiner) according to the usual rules.
Should my child get a chickenpox vaccine?