It looks like the overall goal is:
- For each entry in urls, call$.getand wait for it to complete.
- If it returns just a response without "next", keep that one response
- If it returns a response with a "next," we want to request the "next" as well and then keep both of them.
 
- Call the callback with responsewhen all of the work is done.
I would change #2 so you just return the promise and fulfill it with response.
A key thing about promises is that then returns a new promise, which will be resolved based on what you return: if you return a non-thenable value, the promise is fulfilled with that value; if you return a thenable, the promise is resolved to the thenable you return. That means that if you have a source of promises ($.get, in this case), you almost never need to use new Promise; just use the promises you create with then. (And catch.)
(If the term "thenable" isn't familiar, or you're not clear on the distinction between "fulfill" and "resolve", I go into promise terminology in this post on my blog.)
See comments:
function testCase(urls) {
    // Return a promise that will be settled when the various `$.get` calls are
    // done.
    return Promise.all(urls.map(function(url) {
        // Return a promise for this `$.get`.
        return $.get(url)
            .then(function(response) {
                if (response.meta && response.meta.next) {
                    // This `$.get` has a "next", so return a promise waiting
                    // for the "next" which we ultimately fulfill (via `return`)
                    // with an array with both the original response and the
                    // "next". Note that by returning a thenable, we resolve the
                    // promise created by `then` to the thenable we return.
                    return $.get(url + "&offset=" + response.meta.next)
                        .then(function(nextResponse) {
                            return [response, nextResponse];
                        });
                } else {
                    // This `$.get` didn't have a "next", so resolve this promise
                    // directly (via `return`) with an array (to be consistent
                    // with the above) with just the one response in it. Since
                    // what we're returning isn't thenable, the promise `then`
                    // returns is resolved with it.
                    return [response];
                }
            });
    })).then(function(responses) {
        // `responses` is now an array of arrays, where some of those will be one
        // entry long, and others will be two (original response and next).
        // Flatten it, and return it, which will settle he overall promise with
        // the flattened array.
        var flat = [];
        responses.forEach(function(responseArray) {
            // Push all promises from `responseArray` into `flat`.
            flat.push.apply(flat, responseArray);
        });
        return flat;
    });
}
Note how we never use catch there; we defer error handling to the caller.
Usage:
testCase(["url1", "url2", "etc."])
    .then(function(responses) {
        // Use `responses` here
    })
    .catch(function(error) {
        // Handle error here
    });
The testCase function looks really long, but that's just because of the comments. Here it is without them:
function testCase(urls) {
    return Promise.all(urls.map(function(url) {
        return $.get(url)
            .then(function(response) {
                if (response.meta && response.meta.next) {
                    return $.get(url + "&offset=" + response.meta.next)
                        .then(function(nextResponse) {
                            return [response, nextResponse];
                        });
                } else {
                    return [response];
                }
            });
    })).then(function(responses) {
        var flat = [];
        responses.forEach(function(responseArray) {
            flat.push.apply(flat, responseArray);
        });
        return flat;
    });
}
...and it'd be even more concise if we were using ES2015's arrow functions. :-)
In a comment you've asked:
Could this handle if there was a next next? Like a page 3 of results?
We can do that by encapsulating that logic into a function we use instead of $.get, which we can use recursively:
function getToEnd(url, target, offset) {
    // If we don't have a target array to fill in yet, create it
    if (!target) {
        target = [];
    }
    return $.get(url + (offset ? "&offset=" + offset : ""))
        .then(function(response) {
            target.push(response);
            if (response.meta && response.meta.next) {
                // Keep going, recursively
                return getToEnd(url, target, response.meta.next);
            } else {
                // Done, return the target
                return target;
            }
        });
}
Then our main testCase is simpler:
function testCase(urls) {
    return Promise.all(urls.map(function(url) {
        return getToEnd(url);
    })).then(function(responses) {
        var flat = [];
        responses.forEach(function(responseArray) {
            flat.push.apply(flat, responseArray);
        });
        return flat;
    });
}