Yes, they are the same construction if by that you mean using the same words in the same order. But of course some words have multiple meanings. "Have" can be used with several substantive meanings or, as an auxiliary verb, with a purely syntactic meaning.
The opposition had eggs thrown at the minister
is clearly causative: the experience was suffered by the minister rather than the opposition. So the meaning is that the opposition caused one or more third parties to throw eggs.
The minister had eggs thrown at him
has a very slight ambiguity. The words "at him" clearly show that the experiential meaning applies, but could the meaning include BOTH experiential or causative senses? The form of the words at least seems to allow for a DUAL meaning. It is conceivable that the minister secretly told a few of his supporters to throw eggs at him to garner attention or sympathy, but that possibility, in the absence of any other context, is highly implausible. Moreover, if that was the intended meaning, it was ill expressed. It should have been stated as
The minister had eggs thrown at himself
Here the additional causative sense is made clear by using the reflexive pronoun.
When plausibility suggests a highly likely meaning and the unlikely meaning could and probably would have been expressed in a different way, we interpret potentially ambiguous statements as the likely meaning. Humam beings are very good at such linguistic judgments because so many statements offer the potential for unintended meanings.