You can, but if you do, you will be guilty of writing a "run-on sentence." Run-on sentences, while not grammatically incorrect, are poor writing style as they are hard to read.
I would use three sentences:
Thomas gave his son a newspaper clipping. His son saw (or "noticed") that "don't stop looking" was marked on it. He turned the clipping over, and saw that "Black's phone number" was also marked.
In general, avoid running complete sentences together using and or or. Connect complete sentences only when they need to be put together to convey a single idea or closely related ideas, as in the last sentence in my example.