Can "me, too" be used to reply to a negative statement?

You might occasionally hear "me, too" in this situation, but only in casual conversation. Most native English speakers would say "me neither" -- saying "me too" might even suggest you've mis-heard the speaker in some cases, thinking they'd said "I can understand", rather than "I can't understand".

Aside: You will also hear some people say "me either", more commonly in American English.

No, to my understanding, "me too" would be an unusual response to most negative sentences.

We normally answer with a negative construction when we are confirming negative questions, and with a positive construction when we are in confirming positive questions. This can lead to cases that may seem illogical, but are nonetheless considered correct.

For example, let's say that Person A were to ask, "Aren't you coming?" Person B might respond, "No, I am not coming." This is, strictly speaking, a double negative, but is generally considered the correct form. I'm not sure why the language works this way -- it may be a holdover of negative concord, which was normal in Old and Middle English, but uncommon today.

Back to your question, let's say that Person A were to say, "I can't understand it." Person B, if expressing agreement, might respond, "Me neither." If this were a positive expression, Person A might say, "I understand it," to which Person B might respond, "Me too."