Concerning your first question for the Suzuki groups in characteristic 2, it's helpful to go back to the original announcement by Suzuki: A new type of simple groups of finite order, Proc. Nat. Acad. Sci. USA 46 (1960), no. 6, 868-870 (available through JSTOR and maybe other channels). Note that the order of such a group is $q^2(q-1)(q^2+1)$, with $q= 2^{2n+1}$. Suzuki approached his groups in the setting of 2-transitive permutation groups, but later it was seen that they could be viewed efficiently as groups of Lie type: here one gets a split BN-pair of rank 1 and nice relations with an ambient finite group of type $B_2 = C_2$.
Anyway, Suzuki realized that his groups are simple for $q>2$ and that each has only a few conjugacy classes of maximal (proper) subgroups. You've pointed to the Sylow 2-subgroups of order $q^2$, each of which is the full centralizer of a nontrivial central involution (a unipotent element in Lie theory). In addition there are other types of centralizers, of orders $q-1$ (split "torus"), $q+2r+1$, and $q-2r+1$ with $r=2^n$. Here all centralizers turn out to be nilpotent, which is not too surprising from the Lie-theoretic viewpoint since the BN-pair has rank 1.
Given this much information, it's true that "small" classes (which have "big" centralizers) must consist of involutions here.
In the case of the Ree groups in characteristic 3, contained in groups of Lie type $G_2$, the story is probably similar since you again have a "rank" 1 BN-pair (though I haven't checked all details). Note that the Ree groups in characteristic 2 are more complicated, since they are twisted subgroups of Lie type groups $F_4$ and have "rank" 2.
P.S. Looking at the order of a Ree group and some partial information about centralizers in Ward's 1966 paper here, I'm unsure where the data in the question about involutions fits in. Anyway, there must be fairly complete data about centralizers in the literature. ADDED: For a complete list of subgroups, see Theorem 6.5.5 in The Classification of the Finite Simple Groups, Number 3, by Gorenstein-Lyons-Solomon, AMS 1998. These include all centralizers, and it appears that the largest subgroup in the list is the centralizer of (any) involution. If so, this answers the question for Ree groups too. (One of the two G-L-S sources is a paper by Kleidman here.)