You cannot use Part[X, i, j] as a variable in an equation, because it is a command to extract an element from the value of the symbol X, which has no value right now. As a good variable you can use for example X[i, j] (or, as you did before, Subscript[X,i,j], but that is more complicated to manage). You can then collect the solutions into a separate variable, from which you can extract parts:

Eqn = {};
For[ns = 1, ns <= 2, ns++, 
  For[nx = 1, nx <= 2, nx++, AppendTo[Eqn, 2*ns*X[ns, nx] == nx^2]]];
Eqn
MatrixForm[Table[X[i, j], {i, 1, 2}, {j, 1, 2}]]
First[NSolve[Eqn, Flatten[Table[X[i, j], {i, 1, 2}, {j, 1, 2}]]]]
MatrixForm[Xmatrix = Array[X, {2, 2}] /. %]
Part[Xmatrix, 1, 1]

Thanks, Gianluca. Although, my problem is not resolved yet. I created a simple code that generates the same warning messages:

Eqn = {};
For[ns = 1, ns <= 2, ns++, 
  For[nx = 1, nx <= 2, nx++, 
   AppendTo[Eqn, 2*ns*Part[X, ns, nx] == nx^2]]];
Eqn
Flatten[Table[Part[X, i, j], {i, 1, 2}, {j, 1, 2}]]
NSolve[Eqn, Flatten[Table[Part[X, i, j], {i, 1, 2}, {j, 1, 2}]]]

Although I get the final solution, there are many warning messages which I want to get rid of:

Part::partd: Part specification X[[2,1]] is longer than depth of object. >>

{{X[[1, 1]] -> 0.5, X[[1, 2]] -> 2., X[[2, 1]] -> 0.25, 
  X[[2, 2]] -> 1.}}

Any idea to better code this problem?