What's the definition of equational theory? Why is λ logic free?

It is used a first-order language without quantifiers and it is studied a natural deduction calculus for quantifier-free formulas :

Atomic formulas of the form $(s = t)$ are called equations, and the symbol ‘$=$’ is known as equality or identity.

The natural deduction rules for qf formulas are exactly the same as for $LP$ [propositional logic], except that we also have introduction and elimination rules ($=$I), ($=$E) for equality.

Thus, basically, we can construct derivations "made of" equations.

Added

The theory $\lambda$ [of pure lambda calculus] has as terms the set $\Lambda$ ($\lambda$-terms) built up from variables using application and abstraction. The statements of $\lambda$ are equations between $\lambda$-terms [...].

See page 23 :

Note that $\lambda$ is logic free [emphasis added]: it is an equational theory. Connectives and quantifiers will be used in the informal metalanguage discussing about $\lambda$.

From these quotations, the intent of the author is clear. The mathematical theory of pure lambda calculus is based on a syntax of terms, where the set $\Lambda$ of it is built up form variables ($v_0, v_1, ...$), the parentheses and the abstractor $\lambda$ [see Def.2.1.1, page 22], and the formulas of the calculus are equations between terms of the form $M=N$ where $M,N \in \Lambda$ [see Def.2.1.4, page 23].

The formulas have no logical connectives : $\lnot$, $\land$, $\rightarrow$, nor the "usual" quantifiers : $\forall$, $\exists$; in this sense it is "logic free".

Expressions like :

$M = N \implies N = M$

or :

$\forall M \, (\lambda x.x) M = M$

with connectives and quantifiers, are not formulas of the language, but statements in the meta-language describing the $\lambda$ calculus.