You are probably familiar with the way mathematical theories are formally constructed: we have a collection of mathematical objects or notions that have particular properties and are related in particular ways. Some of these notions are taken as primitives: that is, we don't define them, but only give a list of their properties. For example, the notion of "point" in geometry or the notion of "set" in set theory. Other notions are then defined in terms of the primitive ones. For example, the notion of "circle" in Euclidean geometry, defined as a set of points satisfying a particular property. This process is called the "axiomatization" of a theory.
Let's stress two points about axiomatizations.
First, we often have several different choices of primitive notions. So we can have an axiomatization that uses notions $A$ and $B$ as primitives and notion $C$ as a defined one, and an axiomatization that uses notions $A$ and $C$ as primitives and $B$ as defined. The choice between different axiomatizations is a matter of mathematical economy and subjective taste.
Second, just because one notion is taken as primitive and another as defined, it doesn't mean that the primitive one is more intuitive of the defined one. In fact, sometimes primitive notions are less intuitive than defined ones. They are nevertheless chosen as primitives for reasons of economy – for example, they lead to theorems that have more compact statements or easier proofs, as compared with equivalent theorems stated in terms of another set of primitives.
Now, Truesdell's remarks largely concern the axiomatization of a physical theory – which is also a mathematical theory. We can choose a set of physical notions as primitives, in the mathematical sense; and other as defined in terms of the primitives. The choice is again a matter of mathematical economy, subjective taste, and sometimes pedagogical value. See the examples in Suppes's book (1957).
The fact that a physical notion is chosen as primitive doesn't mean that it's historically older or that it's easier to grasp with intuition or experiment.
Experimentally, the second law concerns cyclic processes and exchanges of work and heat. We could use these notions as primitives and do without speaking about entropy. The notion of entropy appears as a sort of convenient book-keeping device when we do not want to consider cyclic processes (see the discussion in Ricou 1986).
As it turns out, if we use it as a primitive concept (again, in the mathematical axiomatization sense) we obtain a mathematical system that's more compact and doesn't force us to consider cyclic processes. But, as Truesdell remarks and we all know from our studies in school, entropy is much less intuitive than heat, work, and cyclic processes – and these have their own intuitive difficulties already!
In fact, entropy cannot even be measured in general. And, contrary to what many basic thermodynamic courses say, it isn't true that we can define it except for a constant term. It turns out that there are many entropy functions that are equivalent in the description of the thermodynamics of particular systems, and they don't differ by only a constant term. This was pointed out by Day (1977, 1988) and is discussed with examples in Owen's very mathematical textbook (1984) and very neatly in Samohýl & Pekař's book (2014). Only for some systems and processes we have an entropy function defined but by a constant term.
There are some works that try to avoid entropy as a primitive and instead derive it as a defined notion; see e.g. Day (1968), Coleman & Owen (1974), and especially Ricou (1986), the discussion by Serrin (1986), and other contributions to the book edited by Serrin (1986b).
I think that Truesdell's quote must, at least partially, be understood in the axiomatization sense above. Entropy is not an intuitive notion, as he himself says in the quote from hyportnex's answer. But it can be a convenient primitive. When we try to define it, e.g. in microscopic terms, we usually obtain something that can't cover all of its uses in continuum mechanics, or something that turns out to implicitly assume a notion similar to that of entropy, so that the definition is circular.
There is research going on about this, and maybe things will change one day. Jaynes's (1979, 1985) generalized notion of entropy, for example, has many interesting properties shared by the entropy functions used in more general continuum thermomechanics.
References
Coleman, B. D., D. R. Owen (1974): A mathematical foundation for thermodynamics https://doi.org/10.1007/BF00251256
Day, A. W. (1968): Thermodynamics based on a work axiom https://doi.org/10.1007/BF00251512
Day, A. W. (1977): An objection to using entropy as a primitive concept in continuum thermodynamics https://doi.org/10.1007/BF01180089
Day, A. W. (1988): A Commentary on Thermodynamics (Springer).
Jaynes, E. T. (1979): Where do we Stand on Maximum Entropy? http://bayes.wustl.edu/etj/articles/stand.on.entropy.pdf
Jaynes, E. T. (1985): Macroscopic prediction http://bayes.wustl.edu/etj/articles/macroscopic.prediction.pdf
Owen, D. R. (1984): A First Course in the Mathematical Foundations of Thermodynamics (Springer).
Ricou, M. (1986): The laws of thermodynamics for non-cyclic processes, in Serrin (1986b) pp. 85–97.
Samohýl, I., M. Pekař (2014): The Thermodynamics of Linear Fluids and Fluid Mixtures (Springer), especially chap. 2.
Serrin, J. (1986): An outline of thermodynamical structure, in Serrin (1986b) pp. 3–32.
Serrin, J. (ed.) (1986b): New Perspectives in Thermodynamics (Springer).
Suppes, P. (1957): Introduction to Logic (Van Nostrand), especially chap. 12.
