Mathematica is giving you the eigenvalues as the 6 roots of a 6-degree polynomial, which has no closed form solution. You can however define a function which when provided real values for Eg, Ej, and ng does return a numerical result. Note the use of N to get numbers.

In[1]:= matr = {{Ec*(0 - ng)^2, -Ej/2, 0, 0, 0, 0}, {-Ej/2, 
    Ec*(1 - ng)^2, -Ej/2, 0, 0, 0}, {0, -Ej/2, Ec*(2 - ng)^2, -Ej/2, 
    0, 0}, {0, 0, -Ej/2, Ec*(3 - ng)^2, -Ej/2, 0}, {0, 0, 0, -Ej/2, 
    Ec*(4 - ng)^2, -Ej/2}, {0, 0, 0, 0, -Ej/2, Ec*(5 - ng)^2}};

In[2]:= eigenvalues[Ec_, Ej_, ng_] = Eigenvalues[matr];

In[3]:= eigenvalues[2, 3, 6] // N

Out[3]= {1.63846, 8.13635, 18.0644, 32.0357, 50.023, 72.1021}