Help with Voltage Divider driven NPN emitter follower
You are the designer, which means that you are going to choose the NPN transistor to be used. 
As the load is around \$25 mA\$, a small signal model can be used which normally has a \$\beta = 100\$ or more. Take your pick of a transistor with say \$\beta = 150\$.
Good specs give a range of beta, say \$\beta = 100..200\$.
Then there's the \$V_{be}\$ that's of importance, look them up for the transistor you chose and in the operating points of \$0..5V/25 mA\$, assuming the current through \$R_3\$ neglible with respect to \$I_1=25 mA\$.

Now establish 'worst' and 'best' conditions.
It's clear that the output voltage will be highest if \$I_1=0\$ and \$\beta\$ is maximal and \$V_{be}\$ is minimal.
The other way around, \$V_1\$ will be lowest if \$I_1\$ highest, \$\beta\$ is minal and \$V_{be}\$ is maximal.

Now write the output voltage as function of the parameters \$R_1..R_3\$, \$V_{be}\$, \$\beta\$, \$I_1\$: \$U_1 = f (R_1, R_2, R_3, V_{be}, \beta, I_1)\$.
Establish the total differential of this function with respect to \$V_{be},\beta, I\$, like: \$\partial U_1 = \frac{\partial U_1}{\partial V_{be}} . dV_{be} + \frac{\partial U_1}{\partial \beta} . d\beta + \frac{\partial U_1}{\partial I_1} . dI_1\$, 'simplify' to \$\Delta U_1=\frac{\partial U_1}{\partial V_{be}} . \Delta V_{be} + \frac{\partial U_1}{\partial \beta} . \Delta\beta + \frac{\partial U_1}{\partial I_1} . \Delta I_1\$ and put in the \$\Delta\$s the minima and maxima that lead to the worst and best results respectively.  

Now solve \$R_1, R_2, R_3\$ to stay within your desired 95..100% of 5V.
This last step is the tricky part, and that's where electrical engineers developed a preference to a mixture of 'educated guesses' and iteration ('trial and error') that you seem to want to pass-by at the cost of higher efforts in solving sets of mathematical equations.
Actually your approach might be better in very complex situations which most engineers would try to avoid during their design by chopping up the problem is small partial problems, which then can be solved with the methodologies of 'guess and iterate'.