What sort of rights do men have in a civil society according to Edmund Burke in the middle of Reflections on the Revolution in France?

As it unfolded, even before the beheadings, the French Revolution was an unsettling development for supporters of monarchy. While the Enlightenment ideas of individual rights did enjoy widespread support, the imminence of anarchic chaos was alarming. The Anglo-English Edmund Burke, a member of Parliament, was among the most outspoken critics of the "tragi-comic" event. In urging caution and reform over revolution, he criticized not only the French actions but many of the underlying principles.
Burke included substantial discussion of "rights" as they figured in civil society. He wrote that civil society, if it were to be

for the advantage of man, all the advantages for which it is made become his right. It is an institution of beneficence; and law itself is only beneficence acting by a rule.

Concerned for the balance of rules and rights, he advocated the concrete over the abstract.

Burke pointed out that civil society provides certain guarantees, including justice, earnings, inheritance, aid in childrearing, and even consolation in death. However, he opposed the prior definition of an abstract, universally applicable concept of such guarantees. Rather, the specific, concrete ways of acquiring and safeguarding rights was a matter for social experience to be developed gradually through custom and law.

In particular, Burke did not support direct democracy.

[A]s to the share of power, authority, and direction which each individual ought to have in the management of the state, that I must deny to be amongst the direct original rights of man in civil society.

Burke's summation of the way of moderation is one of the most influential passages in the Reflections.

The pretended rights of these theorists are all extremes: and in proportion as they are metaphysically true, they are morally and politically false. The rights of men are in a sort of middle, in incapable of definition, but not impossible to be discerned [emphasis in original].

He advocated for a balance that would avoid extremes, and some would say favored inherited position of the governors—one of the foundation of monarchy—at the expense of the governed.
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