If I'm at the supermarket and want to order a piece of chicken breast, do I say
Ich möchte ein stück Hähnchen bitte.
"Stück" needs to be uppercase because it's a noun, and if you specifically want chicken breast, it's "Hähnchenbrust". Also, the "bitte" should be separated by a comma because it's not an actual part of the main clause. These three are pretty minor mistakes though (two of them inaudible), so your order would be well understood already. This is the correct version:
Ich möchte ein Stück Hähnchenbrust, bitte.
For even more politeness (yours is already polite enough, don't worry), you could also say:
Ich hätte gerne ein Stück Hähnchenbrust, bitte.
If I would sell chicken and parts of chicken, and you say to me:
Ich hätte bitte gerne ein Stück Hähnchen/Huhn.
I'd like to have a piece of chicken.
Then I would ask you:
Was für ein Stück?
Which piece?
Because I would think you don't want a whole chicken, but just a part of it, but you didn't tell me if you want a leg, a wing, a breast or whatever.
If you want a whole chicken, order it like this:
Ich hätte bitte gerne ein Huhn.
I'd like to have a chicken, please.
or, even better, and to make it sure:
Ich hätte bitte gerne ein ganzes Huhn.
I'd like to have a whole chicken, please.
You also can order a half chicken:
Ich hätte bitte gerne ein halbes Huhn.
I'd like to have half of a chicken, please.
If you want one complete breast, order it this way:
Ich hätte bitte gerne eine Hühnerbrust.
I'd like to have one breast of a chicken, please.
Maybe you want just 100 grams of the chicken breast:
Ich hätte bitte gerne 100 Gramm Hühnerbrust.
I'd like to have 100 grams of chicken breast, please.
I give you some more information about Huhn/Hähnchen/Hendl/Güggeli. (You did not ask for it, but maybe you want to know it if you travel to Austria or Switzerland)
The female adult chicken is called »Henne«, and the adult male chicken is called »Hahn«. Since chicken are small animals, you often use a diminutive. But the diminutive is built different in different regions:
Austria and Bavaria:
In Austria, the diminutive is built from the female name, and there is a kind of dimutive that is only used in Austria and Bavaria: You add »erl« to the noun to build the diminutive (Sack -> Sackerl, Krug -> Krügerl, Kind -> Kinderl), and so you get: »Henderl«. But due to sloppy speaking, the e and the r got lost, only the l is left, and so Chicken is in Austria and southern parts of Germany (Bavaria):
das Hendl
Germany, north of Bavaria:
In Germany the diminutive is built from the male form, by adding »chen« to the noun. This a way of building diminutives, that is used everywhere where German is spoken. So in most parts of Germany it is:
das Hähnchen
Switzerland:
In Switzerland you will find a very different word, that also is a diminutive (Swiss and Swabian style diminutive: adding li to the noun), but derived from an old and outdated Swizz word for Chicken (»Gugg«). But this diminutive is only used for roast chicken. It is:
das Güggeli
Note, that all three diminutives are used for male and female chicken, although the Austrian and German variations are derived from words for just one of both genders.
Note also, that all this words are not dialect words, but part of standard German, but with regional usage (there are many of such regional variations in standard German).
Everywhere:
If you want to be understood everywhere, don't use a diminutive for chicken. You'd better use the normal noun for chicken of any biological gender instead:
das Huhn
That depends on what exactly you want to order, I usually go with
Ich hätte gern x Gramm Hähnchenbrust, bitte.