align multiline equations with labels

To achieve the first part of your formatting objective, don't employ three independent align environments. Instead, employ a single align environment. (I must confess to not understanding the part about aligning the \varepsilon symbols.)

A separate comment/suggestion: You should not use \textsubscript to create subscripts in math mode. Instead, please use _.

I must further confess to not being sure what the * (asterisk) symbols are suppose to denote. If they denote multiplication, you should either replace them with \cdot symbols or, better still, just omit them entirely.

\documentclass{article}
\usepackage{mathtools}
\usepackage[noabbrev]{cleveref} % optional

\begin{document}
\Cref{eq:p1_m1} tests one model, \cref{eq:p1_m2} the extended model, and \cref{eq:p1_m3} a third one.
\begin{align}
     ABC_{n}
     &= \alpha_{0} + \alpha_{1} E_{ij} + \alpha_{2} E_{ij} * F_{i} + \varepsilon_{i}
     \label{eq:p1_m1}\\
     ABC_{n}
     &=  \alpha_{0} + \alpha_{1} E_{ij} + \alpha_{2} E_{ij} * F_{i} \nonumber \\
     &\quad + \alpha_{3} \Delta_{G_{ij}} + \varepsilon_{i}
     \label{eq:p1_m2}\\
     ABC_{n}
     &=  \alpha_{0} + \alpha_{1} E_{ij} + \alpha_{2} E_{ij} * F_{i} \nonumber \\
     &\quad + \alpha_{3} \Delta_{G_{ij}} + \alpha_{4} E_{ij} * G_{nt} \nonumber \\
     &\quad + \alpha_{5} E_{ij} * F_{i} * G_{nt} + \alpha_{6} \Delta_{G_{ij}} * G_{nt} + \varepsilon_{i}
     \label{eq:p1_m3}
\end{align}
\end{document}

Here is a proposition with alignat and \mathllap. I also suggest using _the commands from cleveref to save quite some typing with cross-references:

\documentclass{article}
\usepackage{mathtools}
\usepackage{cleveref}
\crefname{equation}{equation}{equations}

\begin{document}

            \Cref{eq:p1_m1} tests one model, \cref{eq:p1_m2,eq:p1_m3} the extended model and a third one respectively.



    \begin{alignat}{2} \label{eq:p1_m1}
                    ABC_{\mathrm{n}} &= \alpha_{0} + \alpha_{1} E_{\mathrm{ij}} + \alpha_{2} E_{\mathrm{ij}} * F_{\mathrm{i}} & &{} + \varepsilon_{\mathrm{i}}\\[1ex]
  \label{eq:p1_m2}
 ABC_{\mathrm{n}} &= \alpha_{0} + \alpha_{1} E_{\mathrm{ij}} + \alpha_{2} E_{\mathrm{ij}} * F_{\mathrm{i}} \notag \\
 & &\mathllap{{}+ \alpha_{3} \Delta_{G_{\mathrm{ij}}}} & + \varepsilon_{\mathrm{i}}\\[1ex]
      \label{eq:p1_m3}
 ABC_{\mathrm{n}} &= \alpha_{0} + \alpha_{1} E_{\mathrm{ij}} + \alpha_{2} E_{\mathrm{ij}} * F_{\mathrm{i}} \notag \\
 &\phantom{{}=} + \alpha_{3} \Delta_{G_{ij}} + \alpha_{4} E_{\mathrm{ij}} * G_{\mathrm{nt}} \notag\\
 & & \mathllap{{}+ \alpha_{5} E_{\mathrm{ij}} * F_{\mathrm{i}} * G_{\mathrm{nt}} + \alpha_{6} \Delta_{\mathrm{G_{\mathrm{ij}}}} * G_{\mathrm{nt}}} & {}+ \varepsilon_{\mathrm{i}}
                \end{alignat}

\end{document}