Parse my Esperanto!

QuadR, 65 bytes

.x
3::⍵M⋄'ĉĝĥĵŝŭĈĜĤĴŜŬ'['cghjsuCGHJSU'⍳⊃⍵M]

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.x replace any char followed by "x" with

3::⍵M upon indexing error, return the match unmodified
⋄ now try:
 'ĉĝĥĵŝŭĈĜĤĴŜŬ'[…] index into this string with
  ⍵M the match's
  ⊃ first letter's
  ⍳ index
  'cghjsuCGHJSU' in this string

This is equivalent to the Dyalog APL tacit function:

'.x'⎕R{3::⍵.Match⋄'ĉĝĥĵŝŭĈĜĤĴŜŬ'['cghjsuCGHJSU'⍳⊃⍵.Match]}

Retina, 27 bytes

iT`x`̂`[cghjs]x
iT`x`̆`ux

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This program is composed by two transliterations. Due to having combining characters in the code this doesn't render too well, the first line should actually look similar to iT`x`^`[cghjs]x, where ^ stands for the circumflex accent combining character. What this is saying is that it should Transliterate (ignoring case) all the xs in the input into a ^, whenever they are following any letter in [cghjs].

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Note: TIO incorrectly measures this code as 25 bytes. Actually, this Retina program uses UTF-8 encoding (other programs can use UTF-32 or ISO 8859-1) and the two combining characters present cost 2 bytes each.

C,  173  154 bytes

Thanks to @Colera Su for saving 17 bytes!

p,c,i;f(char*s){for(char*l="cghjsuCGHJSU";p=*s;~c&&putchar(p))for(c=*++s,i=0;c=='x'&&l[i];++i)l[i]-p||write(1,"ĉĝĥĵŝŭĈĜĤĴŜŬ"+i*2,2,c=-1,++s);}

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Explanation:

p,c,i;
f(char*s)
{
    // The outer loop and an array of characters that are modified by a trailing 'x'.
    // The array/string is used for getting the index for the accented character later.
    for (char*l="cghjsuCGHJSU";

                                // Store the current character of the input string in 'p'.
                                // If it is '\0', the loop terminates.
                                p=*s;

                                      // The last statement in the loop.
                                      // If 'c==-1', it outputs the char stored in 'p'. 
                                      ~c&&putchar(p))

        // Store the character following 'p' in 'c' and increment the string pointer.
        for(c=*++s, i=0;

                        // If 'c' is not the letter 'x', the inner loop terminates
                        // immediately. Otherwise it loops through the characters of
                        // string 'l'.
                        c=='x'&&l[i]; ++i)

            // If the character stored in 'p' is found inside the string 'l'...
            l[i]-p ||

                      // ...then print the accented character corresponding to 'p'.
                      // 'i' is the index of 'p' in 'l', and, because the characters
                      // with accents are two bytes each, the index is multiplied by 2.
                      write(1,"ĉĝĥĵŝŭĈĜĤĴŜŬ"+i*2,2,

                      // Finally set 'c' to -1 so that the non-accented character doesn't
                      // get printed too, and increment the string pointer so that the
                      // letter 'x' doesn't get printed either.
                                                    c=-1, ++s);
}