Quantum Mechanics , bra-ket , angular momentum eigenkets, eigenvalues

I have a question on the algebra involved in bra-ket notation, eigenvalues of $\hat{J}$$_{z}$, $\hat{J}$$^{2}$ and the ladder operators $\hat{J}$$_{\pm}$

The question has asked me to neglect terms from O(ε$^{4}$)

I am using the following eigenvalue, eigenfunction results, where l$jm\rangle$ is a simultaneous eignenket of $\hat{J}$$^{2}$ and $\hat{J}$$_{z}$:

1)$\hat{J}$$^{2}$ $|$$jm\rangle$=j(j+1)ℏ$^{2}$$|$$jm\rangle$
2)$\hat{J}$$_{z}$$|$$jm\rangle$=mℏ$|$$jm\rangle$
3)$\hat{J}$$_{\pm}$$|$$jm\rangle$=$\sqrt{(j∓m)(j±(m+1))}ℏ$$|$$j(m±1)\rangle$

So far the working is:(we are told j is fixed at j=1)

$\langle1m'$$|$ ($\hat{1}$-$\frac{ε}{2ℏ}$ ($\hat{J}$$_{+}$ $-$ $\hat{J}$$_{-}$)$+$ $\frac{ε^{2}}{8ℏ}$( $\hat{J}$$_{+}$$^{2}$$+$$\hat{J}$$_{-}$$^{2}$$–$$2$$\hat{J}$$^{2}$$+$$2$$\hat{J}$$_{z}$$^{2}$)) $|$ $1m\rangle$ = $\langle1m'$ $|$ $1m\rangle$$-$$\frac{ε}{2}$($\sqrt{(1-m)(2+m)}$$\langle1m'$$|$ $1(m+1)\rangle$ $+$ $\sqrt{(1+m)(2-m)}$$\langle1m'$$|$ $1(m-1)\rangle$ $+$$\frac{ε^{2}}{4}$((m$^{2}$-4)$\langle1m'$ $|$ $1m\rangle$ +$\frac{1}{2ℏ^{2}}$$\langle1m'$$|$ $\hat{J}$$_{+}$$^{2}$ $+$ $\hat{J}$$_{-}$$^{2}$$|$$1m\rangle$)

My Questions:

– looking at the $\hat{J}$$_{z}$ operator, when it is squared, this has kept the same eigenkets, but squared the eigenvalues. Is this a general result, for eigenvalues and eigenkets? (I have seen this many times and have not gave it a second thought but see next question).
– Using result 3, i would do the same with $\hat{J}$$_{\pm}$ . However my solution says that terms proportional to ( $\hat{J}$$_{+}$$^{2}$ $+$ $\hat{J}$$_{-}$$^{2}$) should be neglected as they will yield only contributions of O(ε$^{4}$).

So for this term I would get (including the constants it is multiplied by) :
$\frac{ε^{2}}{8ℏ^{2}}$$\langle1m'$$|$ $\hat{J}$$_{+}$$^{2}$ $+$ $\hat{J}$$_{-}$$^{2}$$|$$1m\rangle$ = $\frac{ε^{2}}{8ℏ^{2}}$((1-m)(2+m)ℏ$|$$1(m+1)\rangle$+(1+m)(-m)ℏ$|$$1(m-1)\rangle$

And so I can not see where the extra ε$^{2}$ is coming from such that a ε$^{4}$ is yielded that should be neglected.

Many Thanks to anyone who can help shed some light on this, greatly appeciated !

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