This patch is a simplified version of a patch I have sent upstream.
http://gerrit.rockbox.org/1825

-- Chewi

diff -Naur a/base/ziputil.cpp b/base/ziputil.cpp
--- a/base/ziputil.cpp	2019-11-10 06:45:49.000000000 +0000
+++ b/base/ziputil.cpp	2019-12-04 22:38:34.443112450 +0000
@@ -22,9 +22,9 @@
 #include "progressloggerinterface.h"
 #include "Logger.h"
 
-#include "quazip/quazip.h"
-#include "quazip/quazipfile.h"
-#include "quazip/quazipfileinfo.h"
+#include <quazip5/quazip.h>
+#include <quazip5/quazipfile.h>
+#include <quazip5/quazipfileinfo.h>
 
 
 ZipUtil::ZipUtil(QObject* parent) : ArchiveUtil(parent)
diff -Naur a/base/ziputil.h b/base/ziputil.h
--- a/base/ziputil.h	2019-11-10 06:45:49.000000000 +0000
+++ b/base/ziputil.h	2019-12-04 22:38:34.443112450 +0000
@@ -21,9 +21,9 @@
 
 #include <QtCore>
 #include "archiveutil.h"
-#include "quazip/quazip.h"
-#include "quazip/quazipfile.h"
-#include "quazip/quazipfileinfo.h"
+#include <quazip5/quazip.h>
+#include <quazip5/quazipfile.h>
+#include <quazip5/quazipfileinfo.h>
 
 class ZipUtil : public ArchiveUtil
 {
diff -Naur a/rbutilqt.pro b/rbutilqt.pro
--- a/rbutilqt.pro	2019-11-10 06:45:50.000000000 +0000
+++ b/rbutilqt.pro	2019-12-04 22:43:30.144259710 +0000
@@ -131,7 +131,7 @@
 
 DEPENDPATH = $$INCLUDEPATH
 
-LIBS += -L$$OUT_PWD -L$$MYLIBBUILDDIR
+LIBS += -L$$OUT_PWD -L$$MYLIBBUILDDIR -lquazip5
 # append all RBLIBS to LIBS
 for(rblib, RBLIBS) {
     LIBS += -l$$rblib
@@ -142,7 +142,6 @@
 
 # on win32 libz is linked implicitly.
 !win32 {
-    LIBS += -lz
 }
 
 # Add a (possibly found) libspeex now, don't do this before -lrbspeex!
@@ -271,5 +270,4 @@
 
 # source files are separate.
 include(rbutilqt.pri)
-include(quazip/quazip.pri)
 include(logger/logger.pri)