A new result on the Diophantine equation $x^3 + y^3 +z^3 = 3$

Your phrasing in the second paragraph is a bit odd, I guess you mean that the equation "has no integer solution such that $\frac13(x+y+z)$ is a perfect square" (rather than "if"). If I am stressing this, it is because if you submit a paper to any decent research journal as you plan to, it must look completely professional or you face the risk to be dismissed.

You should probably have someone inside academia have a look at your paper and suggest improvement, if you have any one qualified who would be willing to spend some time helping you. If you are afraid of your ideas being stolen (which is in general not something to be afraid of in such a situation), disclose your work publicly or to several different person, so as to establish your paternity. In some cases it is however better to first get advice before public disclosure, as it might avoid embarassment if there is something very wrong in your work.

Then, to your question: I feel that your work, if right and not already present in one form or another in the literature, might be publishable. However Journal of Number Theory seems too selective for your results. Previous relevant papers are :

Zhou, Guo Fu; Kang, Chi Ding "On the Diophantine equation $\sum_1^m k^n=(m+1)^n$" J. Math. Res. Exposition 3 (1983), no. 4, 47–48.

Sondow, Jonathan; MacMillan, Kieren Reducing the Erdős-Moser equation $1^n+2^n+\dots+k^n=(k+1)^n$ modulo $k$ and $k^2$. Integers 11 (2011), A34, 8 pp.

Moree, Pieter(D-MPI) Moser's mathemagical work on the equation $1^k+2^k+\dots+(m−1)^k=m^k$. Rocky Mountain J. Math. 43 (2013), no. 5, 1707–1737

Gallot, Yves; Moree, Pieter(D-MPI); Zudilin, Wadim(5-NEWC-SMP) The Erdős-Moser equation $1^k+2^k+\dots+(m−1)^k=m^k$ revisited using continued fractions. Math. Comp. 80 (2011), no. 274, 1221–1237.

Urbanowicz, Jerzy(PL-PAN) Remarks on the equation $1^k+2^k+\dots+(x−1)^k=x^k$. Nederl. Akad. Wetensch. Indag. Math. 50 (1988), no. 3, 343–348.

E.g. Integers or the Rocky Mountain journal are usually more accessible than JNT, but I cannot say for sure whether your work would be suitable there, as a general principle and because I know so little about it. Involve might be a good match, and more inclined to handle properly submissions from people outside academia like you.

My opinion is:
Step 1.
Repeat the arguments you use in your proofs again and again.
If you are still sure that you have some new (valid) things to say about the equation then go to
Step 2.
Learn how to use LaTeX ,write your paper and submit it to the arXiv.
If it is submitted, then go to
Step 3.
Choose a journal which has previously published papers related to the equation you mention and submit it for publication.
(Don't choose the best journal but a serious one)

It would be better if you choose a title which won't make the editor think that he will probably waste the time of the referees.
A title "Some notes on the equation $x^3 +y^3+z^3=3$" will be safer.

Good luck!!!