I'm in an argument. To me "are" makes more sense. I understand the rationale for is because it's only one chicken, but chickens itself is plural. Help?
If you're in an argument about this, well really you could invent an argument either way. Either way, the argument you invent would probably sound convincing on the surface. Your argument that you "should" use a singular verb will probably hinge around the fact that "one" is singular. Your argument that you "should" use a plural verb might hinge around, say, the fact that it is ungrammatical to say 'A lot of people has...'.
But linguistically, these arguments are generally spurious. Like the argumentation behind many prescriptive rules, they appear at first glance to work logically because they assume a model of language which is overly simplistic. So if you say, e.g. that it "must" be a singular verb because 'one' is singular, really what you're saying is 'my model/understanding of how language works is too simplistic to take account of the fact that sometimes a superficially singular noun is actually part of the subject of a plural verb'.
If you look at actual usage, you will see that it varies. As a native speaker, my gut instinct would be that the plural is more common in spontaneous/informal speech, and that the singular is essentially a prescriptive invention. But I don't actually have any data to support that. (I'm guessing even some of the small available corpora such as Collins Cobuild might provide some data if anyone has time to look at this.)
Peter Shor shows a case where preference for the singular verb apparently overtakes the plural. However, note that the phrase he has chosen uses ellipsis (it "misses out the noun"). If you do some other Ngram searches including the noun, you can find answers where the plural overtakes the singular (e.g. "one in five people have/has"). I'm not sure that Ngram in this case gives a very reliable overall picture of general usage, though it does clearly indicate that both forms are used.
The other answers contain quite good rationales for the two possible answers. I decided to look at usage. Here is a Google Ngram for the phrases "one in a hundred is" and "one in a hundred are".
You can see that "is" is the overwhelming favorite. Google Ngrams can sometimes be misleading, so I looked at a sample of the sources. There are a few interlopers ("One in a hundred is too high a risk"), but in well over half of the occurrences of "one in a hundred is", the subject is "one" and the verb "is". So usage predominantly favors "is", although "are" is also occasionally seen.
So "one in a hundred is" is the preferred construction. However, since there is a reasonably good argument for "one in a hundred are" being grammatical, and since maybe one in five people currently uses it, if you really want to use it, I'd say go ahead. But don't say, * "out of every hundred, one are ...". That sounds really wrong.
The phrase "one out of a hundred" is an adjective, here, qualifying a noun as a particular type of plural: a statistical likelihood. The subject is not "one", but "chickens". That makes the entire phrase a plural construct.
You can also think of it this way: "one-out-of-a-hundred" doesn't have any meaning unless there are 200 or more instances in question. That implies, here, 2 or more chickens.
Would you write "1% is"? Of course not; "1%" doesn't mean "one", it means "Divide a sample by 100, and that will indicate the number of hypothesized instances likely to occur." "1% of all fatalities are...." is the obvious choice. This is the same exact meaning as "One out of a hundred chickens".
It is for this reason that hyphens were invented. The sentence could be written like this:
"One-out-of-a-hundred chickens are...."
So here, "one out of a hundred" is a dependent adjectival phrase modifying "chickens". In instances where the "chickens" is dropped, that does not mean that "one" becomes the subject, but rather that the subject "chickens" is implied.
Finally, to elaborate my point, you may think of it this way:
"Some chickens are overweight."
"How many chickens are overweight?"
"One out of a hundred chickens are overweight."
"One out of a hundred?"
"Yes -- only one out of a hundred are overweight."
Though here the view is brought forward that "are" is correct or that "is" is correct but "are may be used I dare to say that only "is" is correct grammar and "are" is wrong, even when people sometimes use "are".
The sentence is:
One chicken out of a hundred (chickens) is overweight.
It is clear that one chicken is overweight and not a hundred. And even if some people use "are" because they wrongly make the verb agreement with "chickens" I would not say you can also say "are", simply because it is wrong grammar used by people who have little schooling in grammar.
It can depend on how many chickens there are in all. If you only have 100 chickens in total, then 'One in a hundred is' would logically be the only option as 'one' in this case really means only one chicken.
In most cases, though, we are using 'One in a hundred' to state a proportion. For example 'I have 1500 chickens. One in a hundred (is / are) not laying enough eggs.' In this case, 'are' is fine as the 'one' actually represents 15 chickens, not one. One can also singular though (in the reading part of a Cambridge FCE exam paper it says 'one in ten people doesn't like me').