"There are things I can't give up either/too", which is correct?

It's "too".

It's clear that the speaker B is responding to a remark made by somebody else, A. B said "things", not "them". This shows that B is not referring back to any things that A mentioned, but is giving B's own experience with a perhaps different bunch of things. So it's

1. A: There are [things I can't give up].

B: There are [things I can't give up], too.

There are your things, and there are my things, too. The word "either" would be needed only in a negative context, and there is no negative in a suitable position to make "either" needed. Sure, there's a "n't" in the noun phrase "things I can't give up", but it's not at the head of that phrase.

Compare that with:

2. A: There are things I can't give up. Like snacking, and junk food.

B: I can[not give them up], either.

Where B refers to their own inability to do the same action (namely giving up snacking and junk food).

You can't, and I can't, either. The difference here is that the negative (in "can't") negates the verb in the predicate.

Edit for clarity in the light of what other answers have said. The difference between my two examples is not the semantic difference that in the first the bunches of things might be different and in the second they're the same --- rather, it's the syntactic difference that the first contrasts two noun phrases and the second contrasts two predicates whose head verbs are negated. Some more examples based on those in Tim Pederick's answer:

3. A: There are some people I don't like.

B: There are some people I don't like, too.

4. A: I don't like my neighbours.

B: I don't like my neighbours, either.

Some people, some people. The noun phrases are not negative. Don't like, don't like. The predicates' head verbs are negated. Just to show that noun phrases can in some circumstances produce negative contexts that would demand "either":

5. There are no people I dislike so much that I'd say such a rude remark to their faces. There are no people I dislike so much that I'd post such a remark online about them, either.

No people, no people. Negative noun phrases.

Short answer: “Either” is correct. “Too” might not be wrong, but choose “either” just to be safe.

Long answer:

“Either” is commonly used in place of “too” for negative statements (ones using “not”, or in this case “-n’t”).

He likes the new neighbours. I like them too.

He doesn’t like the new neighbours. I don’t like them either.

I’m not sure I can say that “‘too’ for positives, ‘either’ for negatives” is an actual rule of English, or that it’s a universally (or even widely) observed preference. It’s certainly what I (an Australian English speaker) would use myself, and what I’m used to hearing from others. But I wouldn’t be surprised if this is perfectly acceptable to many native speakers:

He doesn’t like the new neighbours. I don’t like them too.

But even among those who (hypothetically) accept this form, I am sure that “either” would also be acceptable. So that’s the safer choice for you.

I will dare to say that you should not use “either” for positives. This is wrong:

He likes the new neighbours. I like them *either.

Also, when I say “negative”, I specifically mean the grammatical negative, using “not” (or “-n’t”). A sentence that says something negative (in the sense of denying or unfavourable) without “not” should use “too”:

He dislikes the new neighbours. I dislike them too.

Either here is used as an adverb. Many grammar sites point to a guideline that is safe to follow:

The adverb either is used in negative sentences (negative verb) to add an agreeing idea or thought. It usually comes at the end of a sentence or clause.

• So you didn't go to the concert. Well, we didn’t either. Was it good? (eceenglish)

Too is an adverb that means also, as well:

You can use too after affirmative sentences. When the verb is negative, you cannot use “too.” (Englishcurrent)

So the matter is simple: In your sentence, the verb modified by either is negative (can't give up). It would be safe to use too if the verb was affirmative:

There are things I can give up too.

Note: There are few cases, however, when too can follow a negative sentence, and it is used to express not agreement, but the addition of one fact to another. For example:

I am so tired this morning. Danny hasn't appeared at the office, too. I honestly doubt we'll finish this task in time for 4 p.m.

This could be re-written as:

I am so tired this morning and on top of that/and, to add to the problem Danny hasn't appeared at the office. I honestly doubt we'll finish this task in time for 4 p.m.

So unless your statement is part of this kind of structure, [e. g. My wife is becoming distant. There are things I can't give up too. I don't see how we can last much longer.]

it is better to use either.