If you have Global Optimization Toolbox, then use MultiStart to run lsqnonlin several times. MultiStart automatically continues if it encounters NaN or Inf values.
Otherwise, it is not too hard to program up this kind of functionality for yourself. If you have finite bounds on all parameters, you can repeatedly use
x0 = lb + rand(size(lb)).*(ub - lb)
as initial points. You can use a try/catch statement to continue after encountering an error.
MATLAB mathematical toolbox documentation