Since one of my answers has been cited, I'll try to clarify why I suggested using IPOPT instead of MINPACK.
My objections to using MINPACK have nothing to do with the algorithms that MINPACK uses and everything to do with their implementation. My main objection is that the software dates back to 1980, and was last updated in 1999. Jorge Moré is retired; I doubt he or any of the other authors of the software keep tabs on it anymore, and there's no team of people actively supporting it. The only documentation I can find on the software is the original, 1980 Argonne technical report written by Jorge Moré and the other MINPACK authors. (Chapters 1-3 can be found here, and Chapter 4 can be found here.) After searching the MINPACK source code and perusing the documentation (the PDFs are scanned images, and can't be searched), I don't see any options to accommodate constraints. Since the original poster of the nonlinear least-squares problem wanted to solve a constrained nonlinear least-squares problem, MINPACK won't even solve that problem.
If you look at the IPOPT mailing list, some users indicate that performance of the package on nonlinear least squares (NLS) problems is mixed relative to Levenberg-Marquardt algorithms and more specialized NLS algorithms (see here, here, and here). The performance of IPOPT relative to NLS algorithms, is, of course, problem dependent. Given that user feedback, IPOPT seems like a reasonable recommendation relative to NLS algorithms.
However, you make a good point that NLS algorithms should be investigated. I agree. I just think that a package more modern than MINPACK should be used because I believe it will perform better, be more usable, and be supported. Ceres seems like an interesting candidate package, but it can't handle constrained problems right now. TAO would work on box-constrained least-squares problems, although it doesn't implement the classic Levenberg-Marquardt, but instead implements a derivative-free algorithm. A derivative-free algorithm would probably work well for large-scale problems, but I wouldn't use it for small-scale problems. I couldn't find any other packages that inspired a great deal of confidence in their software engineering. For instance, GALAHAD doesn't seem to use version control or any automated testing, at first glance. MINPACK doesn't seem to do those things either. If you have experience with MINPACK or recommendations regarding better software, I'm all ears.
With all of that in mind, getting back to the quote of my comment:
Any system of equations is equivalent to an optimization problem, which is why Newton-based methods in optimization look a lot like Newton-based methods for solving systems of nonlinear equations.
A better comment is probably something to the effect of:
When we want to solve a system of $n$ equations with $n$ unknowns $g(x) = 0$, we can formulate this as a least squares optimization problem. (Paraphrase of last paragraph of p.102 of Nonlinear Programming, 2nd edition, by Dmitri Bertsekas.)
This statement holds even for solving systems of equations under constraints. I don't know of any algorithms that are considered "equation solvers" for the case where there are constraints on the variables. The common approach I know of, perhaps jaundiced by several semesters of optimization courses and research in an optimization lab, is to incorporate the constraints on the system of equations into an optimization formulation. If you were to try to use the constraints in a Newton-Raphson-like scheme for equation solving, you'd probably end up with a projected gradient or projected trust-region method, much like methods used in constrained optimization.