Is "or else" used properly in combining the following sentences?1. A new son or daughter was born. 2. The child's name had to be kept secret. 3. A...

The sentence as you have it is fine. You correctly use the
words, "or else." What you have is a coordinating conjunction "or" and an adverb "else",
which is completely fine. What this does is create an alternative between two
options.

I assume that you are asking this question because
the sentence might sound correct if you remove the word
"else." 

That said, there are other ways to do this. A more
economical way to accomplish this is by using the word "otherwise." If you use this
word, then there is no doubt as to what you are trying to
say. 

Another way to rephrase the sentence is by using the
word "because." If you did this the sentence would
read:

When a new son or daughter was born, the child's name
had to be kept secret, because a stranger might hear the name and use it to bewitch the
child.

The merit of this approach is that you answer the
question "why."